Fuzzy Cognitive Network Process: Comparisons With ... - IEEE Xplore

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 3, JUNE 2014

597

Fuzzy Cognitive Network Process: Comparisons With Fuzzy Analytic Hierarchy Process in New Product Development Strategy Kevin Kam Fung Yuen

Abstract—Fuzzy analytic hierarchy process (F-AHP) has increasingly been applied in many areas. However, as the perception and cognition toward the semantic representation for the linguistic rating scale used by the fuzzy pairwise comparison in F-AHP are still open to discuss, F-AHP very likely produces misapplications. This research proposes the fuzzy cognitive network process (F-CNP) as an ideal alternative to F-AHP. A new product development application using F-AHP is revised using F-CNP. This study shows that the proposed F-CNP yields better results due to the appropriate mathematical definition of the fuzzy paired interval scale for the human perception of paired difference. Index Terms—Fuzzy analytic hierarchy process (F-AHP), fuzzy cognitive network process (F-CNP), fuzzy decision analysis, fuzzy pairwise comparison, fuzzy rating scale.

I. INTRODUCTION HE pairwise comparison concept was originated from the psychological research in [33]. Saaty further developed the concept in a mathematical way and applied it to establish the decision models: analytic hierarchy process (AHP) [30] and analytic network process (ANP) [31]. Fuzzy AHP (F-AHP) applies the fuzzy number for the judgment scale of the pairwise comparison, instead of the crisp number of AHP. The extent analysis method (EAM) [9], which is the most popular F-AHP approach, has been progressively applied in various areas with growing number of applications, for example, rock sciences [1], organization capital measurement [3], quality management [6], work system [12], supplier selection [13], [48], technology management [14], product design [17], medical waste management [18], transportation management [22], job evaluation [23], energy technology [24], brand decision [25], mining method [26], information system [27], [35], capital budgeting problem [32], portfolio selection [34], and hospital site selection [36]. Wang et al. [41] recently showed some examples to indicate incorrectness of the EAM and the revised EAM. Yuen [46] further presented some mistakes for both the EAM [9] and the revised EAM [41], while a current study [48] applied the revised EAM [41], although Wang et al. [41] suggested their proposed modified fuzzy Logarithmic Least Squares Method

T

Manuscript received February 17, 2013; revised April 28, 2013; accepted May 14, 2013. Date of publication June 18, 2013; date of current version May 29, 2014. The author is with the Department of Computer Science and Software Engineering, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TFUZZ.2013.2269150

(LLSM) [38]–[41], which is on the improvement of various established fuzzy LLSM methods [2], [37], [42], as the appropriate alternative for the EAM. According to [16], some approaches such as those in [2] and [37] in which normalized weights derived from estimates based on ratio scales can exhibit an irrational ordering due to the problem of normalization procedure of these methods. Yuen [43]–[45], [47] indicated that the basic numerical definition of the paired ratio scale aij = wi /wj inappropriately represents the human perception or cognition of paired difference. The inappropriate definition of the paired ratio scale for AHP follows the inappropriate F-AHP, as the F-AHP applies the fuzzy number, instead of the crisp number in the AHP, to the paired ratio scale to compare two different objects. Yuen [43]–[45], [47] proposed the primitive cognitive network process (P-CNP) using paired interval (or differential) scale to replace AHP’s paired ratio scale potentially producing misapplications. The proposed fuzzy cognitive network process (F-CNP) is the extension of P-CNP by applying the fuzzy set theory to the cognitive rating scale, pairwise opposite matrix, cognitive prioritization operator, and granular aggregation. The word “cognitive” means that CNP emphasizes that the appropriate scale for human cognition of paired difference should be paired interval scale, rather than the AHP/ANP’s paired ratio scale. For instance, we can more easily give the answer or have less chance of creating errors for the difference of two than the ratio of two. The word “cognitive” may refer to cognitive sciences domain for the paired comparisons, as human cognitions of paired comparisons on the basis of measurement scale, number system, and arithmetic operations are open to discussion. The fuzzy cognitive map [19]–[21], [28], [29], [49], although there is similarity in nomenclature, is a completely different domain from this study. The rest of this paper is organized as follows. Section II presents the overview of the activities and steps in F-CNP. Section III reviews the fuzzy pairwise comparison and indicates the misleading results. On the basis of the flaws, Section III proposes fuzzy paired interval (or differential) scale. On the basis of the fuzzy paired interval scale, Section IV develops the fuzzy pairwise opposite matrix (FPOM) for pairwise comparisons, fuzzy accordance index to check the validity of the FPOM, and fuzzy cognitive prioritization operator (FCPO) to compute the fuzzy utility set from the FPOM. Section V presents the fuzzy aggregation operator to compute the fuzzy decision matrix (FDM) combining the fuzzy utility sets and the corresponding weight set. Section VI demonstrates the usability

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TABLE I RATING SCALE SCHEMAS FOR AHP AND CNP IN CRISP AND FUZZY TYPE

Fig. 1. Estimation for Jason’s height using fuzzy analytic pairwise comparison method.

and validity of the proposed method, with comparisons of F-AHP. Section VII concludes the remarks of this research. II. FUZZY COGNITIVE NETWORK PROCESS The F-CNP includes the following activities: 1) the decision problem such as structural decision criteria, exploration of alternatives, and expert profiles are defined; 2) the computational components including the syntactic form and semantic form for fuzzy cognitive comparison scale, FCPO, and fuzzy aggregation operator are defined; 3) all criteria and alternatives are evaluated as FPOMs; 4) FPOMs are converted to fuzzy utility sets by using the cognitive prioritization operator, and then fuzzy utility sets are converted the relative values if necessary; 5) a FDM or a set of FDMs is formed; 6) the utility values and weights in the decision matrix are aggregated; and 7) the best decision from the fuzzy aggregation results is selected subject to decision attitudes. The details are presented in the following sections. III. SCALES FOR PAIRWISE COMPARISON A. Review of Fuzzy Paired Ratio Scale for Pairwise Comparison “The fundamental scale of the AHP is a scale of absolute numbers used to answer the basic question in all pairwise comparisons: How many times more dominant is one element than the other with respect to a certain criterion or attribute?” [31]. A single number is drawn from the scale (for example, paired ratio scale in Table I) to represent dominance in view of the ratio aij = wi /wj [31]. Justification would be much easier if we use a less precise way of expressing judgments, such as words instead of numbers [15]. For example, with respect to corporate image, alternative A is 3 times more preferable than alternative B. Instead, it can be said that A is “moderately” more preferable than B [15]. Or an alternative expression is that “A dominates B, moderate dominance” means “A dominates B 3 times” [11].

A decision maker may feel more confident using a fuzzy number rather than a crisp number in the pairwise comparison. A fuzzy number (see example in Table I) is drawn to represent the linguistic label of dominance in view of the fuzzy ratio scale ˜ i /w ˜j . “A dominates B, moderate dominance” means “A a ˜ij = w dominates B (2, 3, 4) times,” where (2, 3, 4) is a fuzzy triangular number that the membership of the modal value 3 is equal to one, and the memberships of the support interval boundary values 2 and 4 are equal to zero. The crisp number considering only modal value is the special case of the fuzzy number. However, the fuzzy pairwise reciprocal comparison produces misleading results by considering some simple comparison cases using fuzzy ratio scale to represent the linguistic scales. Suppose that the heights of two persons, e.g., Peter and Jason, are compared. If Peter is 150 cm and Jason is 152 cm, we may say that Jason is slightly taller than Peter by our observation. However, if fuzzy paired ratio scale is applied, this becomes another story. The statement that Jason is slightly taller than Peter will be interpreted as the statement that Jason is (1, 2, 3) times as tall as Peter (see Fig. 1). That means the height of Jason is interpreted as (150, 300, 450) cm, providing that Peter is 150 cm. The interpreted result by fuzzy ratio scale is ridiculous. Moreover, if Peter is 150 cm and Jason is 170 m, by our perception, Jason is much taller than Peter. For numerical representation in fuzzy ratio scale, Jason is (4, 5, 6) or (5, 6, 7) times taller than Peter. For more examples, providing that Peter is 45 kg, Jason will be (45, 90, 135) kg if Jason is slightly heavier than Peter (Jason is 47 kg in fact). Providing that Peter is 10 years old, Jason will be (10, 20, 30) years old if Jason is slightly older than Peter (Jason is 11 in fact). Providing that Peter’s IQ is 120, Jason’s will be (120, 240, 360) if Jason is slightly more intelligent than Peter (Jason’s IQ is 123 in fact). B. Perception of Paired Difference Using Fuzzy Paired Interval Scale The aforementioned cases in the previous subsection show that the fuzzy ratio scales for pairwise comparison do not

YUEN: FUZZY COGNITIVE NETWORK PROCESS

599

aforementioned matrix has the following form: ⎡ (0, 0, 0) ⎢ v2 − v1 ⎢ ⎢ B=⎢ .. . ⎣ Fig. 2. Estimation for Jason’s height using fuzzy cognitive pairwise comparison method.

represent the reality of the perception of linguistic comparison, and usually produce exaggerated results beyond our common sense, e.g., either magnification or minification of the human perception. This research proposes fuzzy cognitive pairwise comparison (FCPC) to address this issue. FCPC is the extension form applying fuzzy number to cognitive pairwise comparison. For the case of height comparison, the statement that Jason is slightly taller than Peter will be interpreted as the statement that Jason is (0, 2.5, 5) cm taller than Peter (see Fig. 2). The height of Jason will be interpreted as (150, 152.5, 155) cm, and the semantic interpretation of fuzzy paired interval scale is much more reasonable than fuzzy paired ratio scale to represent the perception of pair comparisons. The technique can similarly be used for the other cases in the previous subsection.

IV. FUZZY COGNITIVE PAIRWISE COMPARISON A. Fuzzy Pairwise Opposite Matrix On the basis of the fuzzy paired interval scale in the last section, the FPOM is proposed to interpret the individual utilities of the candidates. Fuzzy triangular number is chosen as the fuzzy scale due to its popularity in fuzzy applications. Let an ideal fuzzy utility set be V = { v 1 , . . . , v n }, where the fuzzy utility is of the form v i = (vil , viπ , viu ), and the comparison score in fuzzy number is b ij ∼ = v i − v j . The ideal FPOM is B = [ v i − v j ]. A subjective judgmental FPOM using the fuzzy paired interval scale is B = [ b ij ]. B is determined by B as follows: ⎡ ⎤ v1 − v1 v1 − v2 ... v1 − vn ⎢ ⎥ ⎢ v2 − v1 v2 − v2 ... v2 − vn ⎥ ⎥ B = b ij = ⎢ .. .. .. .. ⎢ ⎥ . . . . ⎣ ⎦ vn − v1 vn − v2 ··· vn − vn ⎡ ⎤ b 11 b 12 . . . b 1n ⎢ ⎥ ⎢ b 21 b 22 . . . b 2n ⎥ ∼ ⎢ ⎥ =⎢ . .. .. ⎥ = [ b ij ] = B. (1) .. . ⎣ . . . . ⎦

b n1

b n2

···

b nn

b ij = blij , bπij , buij = − b j i = −buji , −bπji , −blj i , and for i, j

= 1, . . . , n. If i = j, then b ij = v i − v j = (0, 0, 0). Thus, the

⎡

...

(0, 0, 0) .. .

... .. .

vn − v1

v1 − v2

vn − v2

(0, 0, 0) ⎢ u ⎢ −b12 , −bπ12 , −bl12 ⎢ ∼ =⎢ .. ⎢ . ⎣ u −b1n , −bπ1n , −bl1n

v1 − vn ⎤ ⎥ v2 − vn ⎥ ⎥ .. ⎥ . ⎦

· · · (0, 0, 0) l b12 , bπ12 , bu12

(0, 0, 0) .. .

−bu2n , −bπ2n , −bl2n ⎤ l ... b1n , bπ1n , bu1n l ⎥ ... b2n , bπ2n , bu2n ⎥ ⎥ ⎥ = B. .. .. ⎥ . . ⎦

···

(2)

(0, 0, 0)

B can be decomposed as three matrices as follows: ⎡ 0 ⎢ u ⎢ b21 l B =⎢ ⎢ .. ⎣ . bun 1 ⎡ 0 ⎢ π ⎢ b21 π B =⎢ ⎢ .. ⎣ . bπn 1 ⎡ 0 ⎢ l ⎢ b21 Bu = ⎢ ⎢ .. ⎣ . bln 1

bl12

...

0 .. .

... .. .

bun 2

···

bπ12

...

0 .. .

... .. .

bπn 2

···

bu12

...

0 .. .

... .. .

bln 2

···

bl1n ⎤ ⎡ 0 ⎥ ⎢ bl2n ⎥ ⎢ −bl12 ⎢ .. ⎥ ⎥=⎢ . . ⎦ ⎣ .. 0

−bl1n

⎤ ⎡ 0 ⎥ ⎢ bπ2n ⎥ ⎢ −bπ12 ⎢ .. ⎥ ⎥=⎢ . . ⎦ ⎣ .. bπ1n

0

−bπ1n

bu1n ⎤ ⎡ 0 ⎥ ⎢ bu2n ⎥ ⎢ −bu12 ⎢ .. ⎥ ⎥=⎢ . . ⎦ ⎣ .. 0

−bu1n

bl1n ⎤ ⎥ bl2n ⎥ .. ⎥ ⎥ . ⎦

bl12

...

0 .. .

... .. .

−bl2n

···

0

bπ12

...

bπ1n

0 .. .

... .. .

−bπ2n

···

bu12

...

0 .. .

... .. .

−bu2n

···

(3) ⎤

⎥ bπ2n ⎥ .. ⎥ ⎥ . ⎦ 0

(4) bu1n ⎤ ⎥ bu2n ⎥ .. ⎥ ⎥. . ⎦ 0 (5)

Usually, b ij ∈ B is given during the rating process of the expert in the scale in fuzzy number shown in Table I. The expert only fills the fuzzy upper triangular matrix of the following form: +

B =

b ij (0, 0, 0)

ij B = (0, 0, 0) Otherwise written explicitly as ⎡ (0, 0, 0) (0, 0, 0) . . . ⎢ ⎢ b − (0, 0, 0) . . . 21 ⎢ B =⎢ .. .. .. ⎢ . . . ⎣

⎡

b n1

···

b n2

(0, 0, 0) ⎢ u ⎢ −b12 , −bπ12 , −bl12 ⎢ =⎢ .. ⎢ . ⎣ u −b1n , −bπ1n , −bl1n

(0, 0, 0)

⎥ (0, 0, 0) ⎥ ⎥ ⎥ .. ⎥ . ⎦ (0, 0, 0) (0, 0, 0)

(0, 0, 0) .. . −bu2n , −bπ2n , −bl2n

+

... .. .

(0, 0, 0) ⎤ ⎥ (0, 0, 0) ⎥ ⎥. .. ⎥ . ⎦

···

(0, 0, 0)

...

⎤

(7)

and where κ = (κl , κπ , κu ) is the fuzzy normal ¯π) − κ = (κl , κπ , κu ) = (Max(X utility. By default, ℵ ¯ π ) + δ), δ is the average of the ¯ π ), Max(X δ, Max(X ℵ ℵ differences of the modal values of the adjacent atomic terms, ¯ . ¯ π is the set of the modal values from X and X ℵ ℵ AI is the normalized weighted geometric mean of (AI l , AI π , AI u ), and AI ≥ 0. If AI = 0, then B is perfectly ac cordant. If 0 < AI ≤ 0.1, B is satisfactory. If AI > 0.1, B is unsatisfactory. In a fuzzy POM B = { b ij : b ij = (blij , bπij , buij ); i, j = 1, . . . , n} with the fuzzy utilities v i = (vil , viπ , viu ), i = 1, . . . , n, if {viπ } is derived from the accordant POM {bπij }, then {vil } and {viu } are not derived from the accordant matrices since {blij } and {buij } are not accordant when {bπij } is accordant. This problem is invertible due to the rating scale in fuzzy number. For example, let bπik + bπk j = bπij be preserved, and bπik = bπk j = 2. Thus, bπij = 4. To apply in fuzzy case, let b ik = b k j = (1, 2, 3). Although bπik + bπk j = bπij is still valid, blik + blk j = blij and buik + buk j = buij are not valid. Thus, it fol

lows b ik + b k j = b ij as b ij = (3, 4, 5) = (2, 4, 6) = b ik +

−

[ b ij ] is achieved by B = B + B . For a complete compar

ison of a set of candidates, FPOM needs n (n2−1) ratings. B is validated by the fuzzy accordance index AI or FAI is of the following form: 1 1 1 AI = AI l 4 × (AI π ) 2 × (AI u ) 4 where AI l =

∀i, ∀j ∈ (1, . . . , n)(8)

n n 1 l δ , n2 i=1 j =1 ij

2 1 l l l l T (B + (Bj ) − bij ) , δij = Mean κl i ∀i, ∀j ∈ (1, . . . , n)

b k j . Thus, {vil } and {viu } are not derived from the accordant matrices. The weighted geometric mean in FAI offsets the effect of this discordance. Any one of {blij }, {buij }, and {bπij } is accordant and will produce a fuzzy accordant matrix, i.e., FAI = 0 since 1 1 1 AI = (AI l ) 4 × (AI π ) 2 × (AI u ) 4 . The weighted arithmetic mean operator is not appropriate in such case. B. Fuzzy Cognitive Prioritization Operator After a FPOM is formed, the fuzzy utilities can be computed from the FPOM by a FCPO. Regarding cognitive prioritization in crisp number, two methods are suggested in [43] and [45]: primitive least squares (PLS) [or row average plus the normal Utility (RAU)] and least penalty squares (LPS). The development of the FCPOs of this research is the extension of the two aforementioned operators with fuzzy number. The vector of fuzzy individual utilities V = { v 1 , . . . , v n }, l π u v i = (vi , vi , vi ) can be derived by the fuzzy primitive least


601

squares (FPLS) optimization model which is of the following form: l n n (bij − vil + vjl )2 Min Δ = Min π π π 2 u u u 2 i=1 j =i+1 +(bij −vi +vj ) +(bij −vi +vj ) s.t.

n

vil = nκl

i=1 n

viπ = nκπ

i=1 n

viu = nκu

(9)

⎧ β1 , viπ > vjπ & bπij > 0 ⎪ ⎪ ⎪ ⎪ π π π ⎪ ⎪ ⎨ or vi < vj & bij < 0 π βij = β2 , viπ = vjπ & bπij = 0 , 1 = β1 ≤ β2 ≤ β3 ⎪ ⎪ ⎪ ⎪ or viπ = vjπ & bπij = 0 ⎪ ⎪ ⎩ β3 , otherwise ⎧ β1 , viu > vju & buij > 0 ⎪ ⎪ ⎪ ⎪ u u u ⎪ ⎪ ⎨ or vi < vj & bij < 0 u βij = β2 , viu = vju & buij = 0 , 1 = β1 ≤ β2 ≤ β3 ⎪ ⎪ ⎪ ⎪ or viu = vju & buij = 0 ⎪ ⎪ ⎩ β3 , otherwise

i=1

where n = |{ v i }| is a cardinal number of the fuzzy utility vec tor, (blij , bπij , buij ) ∈ B is a fuzzy entry of B, n κ = (nκl , nκπ , nκu ) is the fuzzy population utility, and κ = (κl , κπ , κu ) is the fuzzy normal utility. By default, κ = (κl , κπ , κu ) = ¯ ℵ ), Max(X ¯ ℵ ) + δ), and δ is the average ¯ ℵ ) − δ, Max(X (Max(X of the difference of the modal values of the adjacent atomic terms. The solution of the closed form of FPLS is the fuzzy row average plus the normal utility (FRAU), and the proof can be referred to in the two theorems of Appendix A and B. FRAU is of the following form, as shown in (10), at the bottom of this page. The time complexity for prioritization of a single row vector is O(n). While there are n rows, the complexity of the aforementioned form is O(n2 ). Regarding the fuzzy least penalty squares (FLPS) operator, the vector of the individual utilities can be derived as follows: +

FLPS(B , κ )

= Min Δ = Min

n n

l [βij · (blij − vil + vjl )2

i=1 j =i+1

+

π βij

·

(bπij

u − viπ + vjπ )2 + βij · (buij − viu + vju )2 ]

where

⎧ β1 , vil > vjl & blij > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ or vil < vjl & blij < 0 ⎪ ⎨ l βij = β2 , vil = vjl & blij = 0 , 1 = β1 ≤ β2 ≤ β3 ⎪ ⎪ ⎪ ⎪ or vil = vjl & blij = 0 ⎪ ⎪ ⎪ ⎩ β3 , otherwise

s.t.

n

vil = nκl

i=1 n

viπ = nκπ

i=1 n

viu = nκu

i=1

vil , viπ , viu

≥ 0, i = 1, 2, . . . , n.

If FAI ≤ 0.1, FRAU is recommended. For one reason, FRAU produces the same or very closed result as FPLS, as FAI is very low. For another reason, its computational effort is the least. Thus, FRAU is more preferable when a FPOM is fuzzy accordant, or satisfactory without violation. On the other hand, FPLS is the basic form for developing the FLPS. In view of the approximate accuracy of the discordant matrix with more contradiction, FLPS is more preferable as it minimizes the summation of the multiples of the contradiction and distance errors. However, FLPS may have some approximation errors since different optimization algorithms implemented by software tools may lead to different approximation errors. The accuracy of FLPS depends on the chosen algorithm to solve the optimization problem. Regarding some fuzzy decision problems, the fuzzy priority vector (or fuzzy normalized utility weighting vector) is denoted by W = {w 1 , . . . , w i , . . . , w n }, where w i = (wil , wiπ , wiu ),

⎛

and the summation of the modal values, wiπ , of W is equal to one, i.e., ni=1 wiπ = 1. Thus, W is said to be a fuzzy normalized priority vector (or a fuzzy priority vector in short). In

⎛ ⎞⎞ ⎤ ⎞ i n 1 ⎥ ⎢ ⎜ vil = ⎝ ⎝ bu + bl ⎠ ⎠ + κ l ⎟ ⎥ ⎟ ⎢ ⎜ n j =1 ij j =i+1 ij ⎥ ⎟ ⎢ ⎜ ⎥ ⎟ ⎢ ⎜ ⎥ ⎟ ⎢ ⎜ ⎞ ⎛ ⎥ ⎟ ⎢ ⎜ n ⎥ ⎟ ⎢ l π u ⎜ 1 π π ⎠ π ⎥. ⎟ ⎜ ⎝ , ∀i ∈ {1, . . . , n} (v + κ F RAU (B, κ ) = ⎢ , v , v )| v = b i ij ⎥ ⎟ ⎢ i i i ⎜ n j =1 ⎥ ⎟ ⎢ ⎜ ⎥ ⎟ ⎢ ⎜ ⎥ ⎟ ⎢ ⎜ ⎛ ⎞ ⎞ ⎛ ⎥ ⎟ ⎢ ⎜ i n ⎥ ⎟ ⎢ ⎜ l u ⎠⎠ u ⎠ ⎦ ⎣ ⎝ vu = ⎝ 1 ⎝ +κ bij + bij i n j =1 j =i+1 ⎡

(11)

⎛

(10)

602


TABLE II FUZZY DECISION MATRIX WITH FUZZY AGGREGATED UTILITIES

order to use the proposed utility weighting vector, the fuzzy individual utility set from the FPOM is rescaled (or normalized) as a fuzzy normalized priority vector by the rescale function, or the normalization function, as follows: ⎫ ⎧ l π u ⎪ ⎪ ⎬ ⎨w i = (wi , wi , wi ) : l π u , W= vi vi vi ⎪ ⎪ , , , ∀i ∈ {1, . . . , n} ⎭ ⎩(wil , wiπ , wiu ) = nκπ nκπ nκπ ¯ ℵ ). in which viπ = nκπ and κπ = Max(X (12) i∈{1,...,n }

V. INFORMATION FUSION An fuzzy decision matrix (see Table II) contains a matrix of fuzzy individual utility {(vkl j , vkπj , vkuj )} with respect to a

tor of weights V = ( v 1 , . . . , v j , . . . , v n ). A fuzzy individual utility (vkl j , vkπj , vkuj ) is a triangular fuzzy number for individual

utility of an alternative T k , k = 1, . . . , m with respect to a cri terion C j , j = 1, . . . , n. A normalized fuzzy triangular weight

j =1

k = 1, . . . , m. (14)

vector of alternatives T = { T k , . . . , T k , . . . , T m } and a vector of criteria C = { c 1 , . . . , c j , . . . , c n } associated with a vec

Fuzzy arithmetic mean (FAM) of the following form is chosen as default aggregation operator since it is used by most decision models due to its computational efficiency and comprehensive simplicity. ⎧ ⎞⎫ ⎛ m 1 l l ⎪ ⎪ l ⎪ ⎪ v = v v ⎪ ⎪ ⎪ ⎪ ⎜ Tk n j =1 k j j ⎟ ⎪ ⎪ ⎪ ⎟⎪ ⎜ ⎪ ⎪ ⎪ ⎪ ⎟ ⎜ ⎪ ⎪ ⎪ ⎟⎪ ⎜ ⎪ ⎪ n ⎬ ⎨ ⎟ ⎜ l ⎜ π 1 π π ⎟ π u vk j vj ⎟ vT k , vT k , vT k : ⎜ vT k = V Tk = n j =1 ⎟⎪ ⎜ ⎪ ⎪ ⎪ ⎟⎪ ⎜ ⎪ ⎪ ⎪ ⎟⎪ ⎜ ⎪ ⎪ ⎪ ⎪ ⎟ ⎜ ⎪ m ⎪ ⎪ ⎪ 1 u u ⎠⎪ ⎝ u ⎪ ⎪ ⎪ ⎪ v = v v ⎭ ⎩ Tk kj j n

v j = (vjl , vjπ , vju ) is assigned to a criterion C j , j = 1, . . . , n. A

fuzzy aggregated utility V T k is aggregated by an aggregation operator Agg(•) in the following form:

V T k = (vTl k , vTπ k , vTu k ), ⎧ ⎞⎫ ⎛ l vT k = Agg({vkl j }, vju ) ⎪ ⎪ ⎨ ⎬ ⎟ ⎜ V T k = (vTl k , vTπ k , vTu k ) : ⎝ vTπ k = Agg({vkπj }, vjπ ) ⎠ ⎪ ⎪ ⎩ ⎭ vTu k = Agg({vkuj }, vju ) k = 1, . . . , m. (13) A fuzzy utility (v l , v π , v u ) can be normalized as a fuzzy relative utility (wL , wπ , wu ) using(12) such that nj=1 wπ = 1. Although the relative values have to be used in F-AHP, F-CNP does not always use the relative values for the fuzzy individual utility set. For one thing, the relative utility is a special case of absolute utility. For the other, if the number of alternatives increases, the average utility of each alternative decreases, and finally, the aggregation results of the alternatives are very close and lead to difficulty in justifying the dominate advantage of the best alternative. If the number of alternatives or criteria is not high (i.e., less than six), normalization can be used.

VI. APPLICATION TO NEW PRODUCT DEVELOPMENT To best demonstrate the validity of FCNP, a decision problem in new product development (NPD) [38] is selected for revision and discussion with the comparisons between F-AHP and FCNP. [38] is selected as it includes detailed inputs, methods, procedures, results, and comparisons. The proposed approach in [38], modified fuzzy LLSM [40], is developed from the studies [2], [37], and [42] and is demonstrated to perform better than the most popular F-AHP approach, EAM [9], which has been shown to produce misapplications in [41] and [46]. The objective of this case [38] is to choose one project for the NPD strategy from three candidates. Table III shows the definitions of the criteria and subcrtiteria for measuring NPD projects [5], [10], [38]. The definitions for the rating scales of FCNP and F-AHP are presented in Table IV, respectively. While two objects are compared, forward comparison means that the utility of the former object is over the utility of the latter one, and backward comparison means otherwise. The inputs adapted and transformed from [38] are shown in Tables V–VIII. A. Fuzzy Cognitive Network Process Approach To apply F-CNP, the linguistic representations for the paired rating scale are the same as F-AHP, but the numerical representations of the linguistic terms are different, and are demonstrated in Table IV. Table IV is for the scale conversion purpose of the comparison study. The better definition for the scale of the FCNP should use Table I. The fuzzy accordance index, fuzzy weights, and the related fuzzy utilities of the subcriteria are


603

TABLE III DEFINITIONS OF SCREEN CRITERIA AND SUBCRITERIA FOR MEASURING NPD PROJECTS [5], [10], [38]

where Ωw = {W = (w1 , . . . , wn )|wjU ≥ wjπ ≥ wjL , nj=1 wj = 1, j = 1, . . . , n} is the space of weights, (wjL , wjπ , wjU ) is the normalized triangular fuzzy weight of criterion j (j = 1, . . . , n), and (wkLj , wkπj , wkUj ) is the normalized triangular fuzzy weight of alternative Tk with respect to the criterion j (k = 1, . . . , m; j = 1, . . . , n). However, this form may not be helpful, which will be discussed in the next section. In Table XI, the fuzzy score differences among three projects in F-AHP are wider than the differences in F-CNP shown in Table X, although the ranks are the same as F-CNP. Although there is concept of consistence index in AHP, it seems the concept of the fuzzy consistence index in F-AHP is lacking. Some discussion can be found in [43]. C. Discussions

shown in Table IX. Since all values of fuzzy accordance indices of FPOMs are less than 0.1, all FPOMs are valid. Thus, the data transformed from [38] are suitable for the usability of the F-CNP. The FRAU is chosen as the FCPO, as all FAI values of FPOMs are within acceptable range. The overall synthesis results using the F-CNP are shown in Table X. The FAM is chosen as the fuzzy aggregation operator, due to its computational efficiency, comprehensive simplicity for decision maker, and general acceptance by many decision models. If low fuzzy boundary values of fuzzy scores are taken to be compared, the rank is Project C > Project B > Project A. If up fuzzy boundary or modal values are taken, the rank is Project A > Project B > Project C. Centroid defuzzification may be used to address this contradiction issue of using fuzzy number as result. The overall results of three projects are very close, although Project A is slightly more preferable than Project B, which is slightly more preferable than Project C. B. Fuzzy Analytic Hierarchy Process Approach The computational details of F-AHP can be referred to in [38]–[41]. The synthesis results are presented in Table XI. Wang and Chin [38] used the linear programming (LP) model [4] as the aggregation operator for the fuzzy decision matrix. The argument is that the LP model can produce narrower support intervals [38]. The LP model has the following forms: wTπ k =

n

π π wij wj , k = 1, . . . , m

j =1

wTLk = Min

w ∈Ω w

wTUk = Max w ∈Ω w

n

wkLj wj , k = 1, . . . , m

j =1 n j =1

wkUj wj , k = 1, . . . , m

(15)

By comparing results in Tables X and XI, both F-AHP and F-CNP have the same rank for the candidates. Both methods, however, have different prioritization values for the same linguistic comparison matrix with different numerical representations, and lead to completely different final aggregation values. The aggregation values among three projects by F-CNP are closer to each other than F-AHP, since the paired ratio scale usually exaggerates the perception of the paired difference, and results in wider variance. That is, the lowest one has much lower value, while the highest one has much higher value. Thus, the fuzzy paired ratio scale of the F-AHP to represent the fuzzy difference of two objects is dubious. Regarding aggregation operator, this research suggests the native FAM, while Wang and Chin [38] suggested LP. By comparing results in Tables X and XII, LP model can produce narrower support intervals, and the ranks using LP on the basis of up boundary, low boundary, modal values, and centroid defuzzification are the same as the ranks using the FAM. Such rank observations, however, do not always be guaranteed (see Table XIII). The ideal AO should be easily understood by decision makers, and have less computational complexity. The LP model, therefore, is not recommended, while both produce similar accuracy of results, and the LP model does not show any advantages over the FAM. As aggregation operator is the controversial topic, it is not the main scope of this research, and could be discussed in future study. The implication of F-CNP is that it allows fuzzy inputs and produces fuzzy outputs. Imprecise inputs using fuzzy linguistic terms thus reduce the difficulty of the rating decision, as fuzzy input is easier to make judgment than crisp input. The fuzzy outputs support the decision makers making the optimistic, neutral, and pessimistic decisions, as a fuzzy number shows the range with membership. The up boundary, modal value, and low boundary values of the fuzzy number can represent optimistic, neutral, and pessimistic decisions, respectively. According to the results (see Tables X–XII), the defuzzification step could be redundant, as modal value can be used. Thus, it is not necessary to use centroid defuzzification to produce the crisp value for the final decision. From the scenario proposed by Wang and Chin [38], the same rank produced by both methods could be an occasional

604


TABLE IV CONVERSION TABLE FOR SCALE SCHEMAS OF F-AHP AND F-CNP

TABLE V FUZZY COMPARISON MATRIX FOR FUZZY WEIGHT

TABLE VI FUZZY COMPARISON MATRICES WITH RESPECT TO SUBCRITERIA OF MKTFIT


605

TABLE VII FUZZY COMPARISON MATRICES WITH RESPECT TO SUBCRITERIA OF MANUFIT

TABLE VIII FUZZY COMPARISON MATRICES WITH RESPECT TO SUBCRITERIA OF CUSTFIT, FINRISK, AND UNCERT

case. If only one comparison matrix is changed, overall results using both methods are changed accordingly. For example, B 0 is changed to B 0 as follows: ⎡ ¯ 0 ⎢¯ − ⎢2 ⎢ ¯− B 0 = ⎢ ⎢3 ⎢ ¯− ⎣4 ¯ 2+

¯ 2+ ¯0

¯ 3+ ¯ 2+

¯ 4+ ¯ 3+

¯ 2− ¯ 3−

¯ 0 ¯ 2−

¯ 2+ ¯ 0

¯ 5+

¯ 4+

¯ 5+

⎤ ¯ 2− ¯ 5− ⎥ ⎥ ⎥ −⎥ ¯ 4 ⎥, AI = 0.087. ⎥ ¯ 5− ⎦ ¯ 0

The new fuzzy weight set of B 0 using FCPO is {(0.18, 0.225, 0.27), (0.155, 0.19, 0.225), (0.15, 0.175, 0.2), (0.135, 0.15, 0.165), (0.255, 0.26, 0.265)}. The fuzzy accordance in dex AI is equal to 0.087 and within acceptable range. The new fuzzy weight set of B 0 using fuzzy analytic prioritization operator is {(0.193, 0.2512, 0.3067), (0.1103, 0.1409, 0.1681),

(0.0747, 0.0938, 0.1143), (0.0386, 0.0527, 0.0738), (0.4044, 0.4614, 0.4614)}. The overall results are presented in Table XIII. F-CNP and F-AHP produce different ranks, when only one comparison matrix is changed. F-CNP suggests Project A, while F-AHP suggests Project C. After centroid defuzzification, unlike the former case, both LP and FAM have different ranks, although aggregation variances are very small. FAM is suggested as the reasons have been stated earlier. In view of the hybrid approaches, the fuzzy CNP could be used as rating interface, survey format, or/and weight determination in fuzzy regression [7], fuzzy data compression [8], or/and fuzzy clustering [8] to extend the current application in NPD. VII. CONCLUSION F-AHP has growing applications in many areas. This research indicates that fuzzy paired ratio scale can exaggerate the real paired difference. Many F-AHP applications, nevertheless, do

606


TABLE IX FUZZY ACCORDANCE INDICES, FUZZY WEIGHTS, AND UTILITIES

TABLE X OVERALL SYNTHESIS RESULTS USING FUZZY ARITHMETIC MEAN (FAM) FOR F-CNP

not pay attention to the probably incorrect evaluation resulting from fuzzy paired ratio scale, and very likely produce misleading decisions. This research proposes the F-CNP to address this issue. A number of novel methods are proposed in research. F-CNP applies fuzzy paired interval scale to construct the FPOM, which is validated by the fuzzy accordance index and computed to fuzzy utilities by the FCPO. After the FDM is formed, it is further aggregated to the fuzzy results by the fuzzy aggregation operator. The decision is made on the basis of the results.

The application to NPD successfully demonstrates that F-CNP can provide very reliable decision supports to the NPD project selection as compared with F-AHP. The F-CNP can be the ideal alternative to the F-AHP to be applied to many application domains such as environmental management, energy efficiency decision, resources management, transportation management, business research, information system, and engineering management. Future studies will investigate the extension of type-2 fuzzy sets to CNP, and the other hybrid methods with F-CNP such as TOPSIS, ELECTRE, PROMETHEE, quality


607

TABLE XI OVERALL SYNTHESIS RESULTS USING F-AHP

TABLE XII OVERALL SYNTHESIS RESULTS USING LINEAR PROGRAMMING MODEL FOR F-CNP

TABLE XIII OVERALL SYNTHESIS RESULTS RESULTING FROM CHANGING ONE FUZZY COMPARISON MATRIX

function deployment, clustering, neural network, genetic algorithm, or data envelope analysis. APPENDIX A THEOREM 1 (CLOSED FORM OF PLS) The closed form solution of the PLS optimization model is the RAU, which is of the following form: ⎞ ⎛ n 1 (A1) bij ⎠ + κ ∀i ∈ {1, . . . , n} . vi = ⎝ n j =1

Proof: To have the closed form solution, the partial differen¯ with respective to all vk ∈ V is derived and is of the tiation of Δ ¯ form: Lk = δδvΔk = 0, k = 1, 2, . . . , n. Then, the linear system {Li } is solved for V . Therefore ¯ = Δ

n n

(bij − vi + vj )2

i=1 j =i+1

¯ = Δ

i= k j > i,j = k

Δij +

i=k ,j > i,j = k

Δk j +

i= k ,j > i,j =k

Δik

608


¯ = Δ

Δij +

i= k j > i,j = k

¯1 = Let Δ

n ≥j > k

Δik .

i< k

¯2 = Δij , Δ

i= k j > i,j = k

¯3 = Δ

Δk j +

Δk j ,

and

n ≥j > k

Δik .

i< k

Thus ¯ =Δ ¯1 + Δ ¯2 + Δ ¯3 Δ ¯ ¯1 ¯2 ¯3 δΔ δΔ δΔ δΔ = + + δvk δvk δvk δvk Case 1:

¯1 δΔ δvk ¯2 δΔ δvk

=

δ(

i = k

j > i , j = k

Δij )

δvk δ ( n ≥j > k Δ k j ) δvk

δ(

= = Case 2: −2 n ≥j > k (bk j − vk + vj ). δ( i< k Δik ) ¯ Case 3: δδΔv k3 = = δvk 2 i< k (bik − vi + vk ). To combine the three cases

= 0.

n ≥j > k

δ(

i< k

(b k j −v k +v j ) 2 ) δvk

=

(b i k −v i +v k ) 2 ) δvk

=

¯ ¯1 ¯2 ¯3 δΔ δΔ δΔ δΔ = + + δvk δvk δvk δvk = −2 (bk j − vk + vj ) + 2 (bik − vi + vk ) n ≥j > i

⎛

i< k

= −2 ⎝

(bk j ) − (n − k) vk +

n ≥j > k

+2

(bik ) −

(bk j ) − (n − k) vk +

n ≥j > k

+2 ⎛

(bik ) −

i< k

+⎝

(vi ) + (k − 1) vk

i< k

⎛

⎛

(bik ) −

⎛ + ⎝−

i< k

⎞⎞

⎛

(bk i ) −

⎞ (vi )⎠

1≤i< k

(bk j )⎠⎠

n ≥j > k

(vj ) +

n ≥j > k

1≤i< k

= 2 ⎝((n − 1) vk ) − ⎝

⎛

n ≥j > k

(vj )⎠

n ≥j > k

= 2 ⎝((n − 1) vk ) − ⎝ ⎛

⎞

(vj ) + ⎞⎞

⎞

To divide the aforementioned system by n, the reduced row echelon form is ⎤ ⎡ 1 1 0 0 ...0 (b1i ) + κ n i ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎢0 1 0 ...0 (b2i ) + κ ⎥ ⎥ ⎢ n ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ [I V ] = ⎢ 1 ⎥ ⎢0 0 1 ...0 (b3i ) + κ ⎥ ⎥ ⎢ n ⎥ ⎢ i ⎥ ⎢. . . . .. ⎥ ⎢. . . . . . . ⎥ ⎢. . ⎥ ⎢ ⎦ ⎣ 1 (bn i ) + κ 0 0 ... ...1 n i Interestingly, the result is the same as the row average plus the normal utility.

(vi )⎠

APPENDIX B THEOREM 2 (CLOSED FORM OF FPLS)

i< k

(bk j )⎠⎠

n ≥j > k

⎛ ⎞ ⎛ ⎞⎞ = 2 ⎝((n − 1) vk ) − ⎝ (vi )⎠ − ⎝ (bk j )⎠⎠ . i= k

V is solved by Gaussian elimination of [ E b ], in which the final row is added to other rows. Thus ⎤ ⎡ n 0 0 . . . 0 nκ + (b1i ) ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ ⎢ 0 n 0 . . . 0 nκ + (b2i ) ⎥ ⎥ ⎢ ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ . ⎢ 0 0 n . . . 0 nκ + (b3i ) ⎥ ⎥ ⎢ ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ .. .. . . .. . . ⎥ ⎢. . . . . ⎥ ⎢ ⎦ ⎣ (bn i ) 0 0 . . . . . . n nκ + i

(vi ) + (k − 1) vk

i< k

= −2 ⎝

(vj )⎠

n ≥j > k

i< k

⎛

⎞

¯ ¯ Since ∂ 2 Δ/∂v k = 2 (n − 1) > 0, Δ is convex. As Lk = ¯ δ Δ/δv = 0, k = 1, 2, . . . , n, there exists a minimal vk , k = k 1, 2, . . . , n. The values can be solved by a linear system. Thus, the augmented matrix [ E b ] is of the following form: ⎤ ⎡ n−1 −1 −1 ... − 1 (b1i ) ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ ⎢ −1 n−1 −1 ... − 1 (b2i ) ⎥ ⎥ ⎢ ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (b3i ) ⎥ −1 −1 n−1 ... − 1 ⎢ [ E b ] =⎢ ⎥. i ⎥ ⎢ ⎥ ⎢ . . . .. .. ⎥ ⎢ .. .. . . . . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (bn i ) ⎥ −1 ... ...n − 1 ⎢ −1 ⎥ ⎢ i ⎦ ⎣ 1 1 ··· ···1 nκ

j

The solution of the closed form of FPLS is the FRAU in (10). Proof: As Δl , Δπ , and Δu are independent, FPLS can be transformed into multiple objective programming

MPLS(B + , κ ) = Min Δl =

n n l 2 bij − vil + vjl i=1 j =i+1


609

n n π 2 bij − viπ + vjπ

Min Δπ =

i=1 j =i+1

Min Δu =

n n u 2 bij − viu + vju i=1 j =i+1

n

s.t.

vil = nκl

n

i=1 n

viπ = nκπ

i=1

viu = nκu .

i=1

The multiple objective programming can be transformed into three optimization models as follows:

FPLS(B + , κ ) = (FPLSl , FPLSπ , FPLSu ), where +

FPLSl (B l , κl ) = Min

l

Δ=

n n

(blij − vil + vjl )2

i=1 j =i+1 n

s.t.

vil = nκl .

i=1 +

FPLSπ (B π , κπ ) = Min

π

Δ =

n n

(bπij − viπ + vjπ )2

i=1 j =i+1 n

s.t.

viπ = nκπ .

i=1 +

FPLSu (B u , κu ) = Min

u

Δ =

n n

(buij − viu + vju )2

i=1 j =i+1

s.t.

n

viu = nκu .

i=1

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Kevin Kam Fung Yuen received the B.Sc. (Hons.) degree in enterprise engineering and e-business and the Ph.D. degree in computational intelligence and operations research from the Hong Kong Polytechnic University, Hung Hom, Hong Kong, in 2004 and 2009, respectively. He is currently a Lecturer with the Department of Computer Science and Software Engineering, Xi’an Jiaotong-Liverpool University, Suzhau, China. Previously he was an Assistant Professor with the Faculty of Economics and Administrative Sciences, Zirve University, Gaziantep, Turkey. His research interests include computational intelligence, decisions analysis, information systems, algorithms, social network, and operations research. He has published more than 35 research articles in the journals, book chapters, and conferences. He serves as a Reviewer for various journals and conferences, including more than 20 SCI/SSCI journals.